Preface If we wish to foresee the future of mathematics our proper course is to study the history and present condition of the HENRI POINCARE This book treats the major mathematical creations and developments from ancient times through the first few decades of the twentieth century. It aims to present the central ideas,with particular emphasis on those currents of activity that have loomed largest in the main periods of the life of mathematics and have been influential in promoting and shaping sub- sequent mathematical activity.The very concept of mathematics,the changes in that concept in different periods,and the mathematicians'own understanding of what they were achieving have also been vital concerns. This work must be regarded as a survey of the history.When one considers that Euler's works fill some seventy volumes,Cauchy's twenty-six volumes,and Gauss's twelve volumes,one can readily appreciate that a one-volume work cannot present a full account.Some chapters of this work present only samples of what has been created in the areas involved,though I trust that these samples are the most representative ones.Moreover,in citing theorems or results,I have often omitted minor conditions required for strict correctness in order to keep the main ideas in focus.Restricted as this work may be,I believe that some perspective on the entire history has been presented. The book's organization emphasizes the leading mathematical themes rather than the men.Every branch of mathematics bears the stamp of its founders,and great men have played decisive roles in determining the course of mathematics.But it is their ideas that have been featured;biography is entirely subordinate.In this respect,I have followed the advice of Pascal: "When we cite authors we cite their demonstrations,not their names." To achieve coherence,particularly in the period after 1700,I have treated each development at that stage where it became mature,prominent, and influential in the mathematical realm.Thus non-Euclidean geometry is presented in the nineteenth century even though the history of the efforts to